Curved Yang-Mills-Higgs theory
We will discuss how to define gauge theories making use of principal bundles $P$ equipped with a Lie groupoid $G$ action instead of a Lie group action. In order to define Ehresmann connections on $P$ we will fix a connection on $G$, a generalisation of the concept of the Maurer-Cartan connection. In particular, as I pointed out in my Ph.D. thesis, the connection on $G$ should be a closed form and its curvature exact, w.r.t. a natural simplicial differential. We will see that the closedness is needed for the kinematics and the exactness for the dynamics and gauge invariance of the associated gauge theory. This is partially a joint work with Mehran Jalali Farahani, Hyungrok Kim, and Christian Sämann. The presented examples of curved Yang-Mills-Higgs theories are results directly inherited by my joint work with Camille Laurent-Gengoux.
Quantization and Reduction for CR Line Bundles
We consider a compact, torsion-free CR manifold that admits an action by a compact Lie group. Given an equivariant rigid CR line bundle, it is natural to study the space of invariant CR sections, onto which a particular weighted invariant Fourier–Szegő operator projects. Under natural assumptions, we show that the group-invariant Fourier–Szegő projector admits a full asymptotic expansion. As an application, if the tensor power of the line bundle is sufficiently large, we prove that quantization commutes with reduction. This is joint work with Chin-Yu Hsiao.
The tt*-Toda equations and symplectic groupoids (2)
We present some of the background to - and wider aspects of - our results on the symplectic geometry of the tt*-Toda equations which were described by Nan-Kuo Ho in her talk on our joint project. As she explained, the tt*-Toda equations describe isomonodromic families of certain meromorphic connections on the Riemann sphere. They are a generalization of the Third Painleve Equation, and a special case of the cyclic harmonic bundle (or Higgs bundle) equations. They were introduced by the physicists Cecotti and Vafa in the 1990's as a way to describe (and possibly classify) certain field theories.
First we discuss the relation with Painleve theory, which provides some general intuition regarding the various spaces and the source of their symplectic structure. Then we explain how special types of solution are related to geometry of harmonic bundles and Frobenius manifolds, a point of view due to Dubrovin. Finally we describe some further aspects of our particular example, which is a symplectic groupoid structure on a certain moduli space of solutions of the tt*-Toda equations, including a relation with Bondal's groupoid of semi-orthogonal bases.
The tt*-Toda equations and symplectic groupoids (1)
Via Riemann Hilbert correspondence, meromorphic connections are determined by their monodromy data. In this talk, we will look at the meromorphic connections related to the tt*-Toda equations. We will explain how the monodromy data corresponding to the tt*-Toda equations form a nice symplectic groupoid. This is a joint work with Martin Guest.
Classification of neighborhoods fo leaves of singular foliations
Classifying singular foliations seems hopeless, even in a neighborhood of a leaf. However, it turns out that there are significative less singular foliation having a given leaf and a given transverse structure than one could expect. We will also give several conjectures about more involved structures, like Lie algebroids or Poisson brackets.
This is a joint work with Simon Raphaël Fischer, and a prolongation of some results of his PhD about Yang-Mills bundles, i.e. Lie group bundles with a curvature which is inner.
Deformation quantization of singular phase spaces by homological reduction
In the first part of the talk we explain the method of homological reduction and its quantized version. We then show that quantized homological reduction can be applied to construct deformation quantizations of certain singular symplectic spaces, namely singular symplectic quotients, where the coefficients of the moment map define a complete intersection. In the second part of the talk examples are discussed, among others one where the singularity type is worse than an orbifold singularity and a lattice gauge model. We also present a few new results on equivariant analytic structures on symplectic manifolds and the analyticity of the moment map.
Pairs of spectral projections of quantum observables on Riemann surfaces
We discuss the semiclassical behaviour of pairs of spectral projections corresponding to incompatible quantum observables, in the framework of geometric quantization of closed Riemann surfaces.
Noncommutative calculi for differential graded manifolds
Differential graded manifolds are a useful concept that provides a common framework for many important classes of algebro-geometric structures, including homotopy Lie algebras, foliations, and complex manifolds. In this talk, we will describe noncommutative calculi for differential graded manifolds and then outline a Duflo-Kontsevich type theorem. The Duflo theorem of classical Lie theory and the Kontsevich theorem regarding the Hochschild cohomology of complex manifolds are special instances of this theorem.
BRST, BV, and the Noether theorems
BRST is a procedure of quantising theories with gauge redundancies and BV is a more general formalism allowing non-closure of the gauge algebras off-shell. In this talk, we explore the BRST current, its relation with Noether's first and second theorems in classical field theory, and the role in asymptotic symmetry. This is based on a joint work with Baulieu and Wetzstein.
$BV_\infty$ quantization of (-1)-shifted derived Poisson manifolds
In this talk, we will give an overview of (-1)-shifted derived Poisson manifolds in the $C^\infty$-context, and discuss the quantization problem. We describe the obstruction theory and prove that the linear (-1)-shifted derived Poisson manifold associated to any $L_\infty$-algebroid admits a canonical $BV_\infty$ quantization. This is a joint work with Kai Behrend and Matt Peddie.